#8496523

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4352 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,4,5)(3,7,8,6)(12,20)(16,21)(19,24)(22,23)(26,32)(28,36)(29,37)(33,40);(2,5)(6,7)(9,10)(11,15)(12,21)(13,17)(14,18)(16,20)(19,22)(23,24)(25,26)(27,33)(28,34)(29,38)(30,36)(31,32)(35,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,19)(14,17)(15,18)(16,22)(20,24)(21,23)(25,27,31,39)(26,33,32,40)(28,37,36,29)(30,38,34,35);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,24)(21,22)(25,28)(26,34)(27,37)(29,39)(30,32)(31,36)(33,35)(38,40);(1,3,4,8)(2,6,5,7)(9,12)(10,16)(11,19)(13,22)(14,20)(15,23)(17,24)(18,21)(25,29,31,37)(26,35,32,38)(27,28,39,36)(30,40,34,33);(9,13,14,15)(10,17,18,11)(12,22,20,23)(16,24,21,19)(25,30,31,34)(26,36,32,28)(27,38,39,35)(29,40,37,33)

Polycyclic presentation:

2^9|a^2=h,c^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8403719);